Abstract
In this note, we study the general form of a multiplicative bijection on several families of functions defined on manifolds, both real or complex valued. In the real case, we prove that it is essentially defined by a composition with a diffeomorphism of the underlying manifold (with a bit more freedom in families of continuous functions). Our results in the real case are mostly simple extensions of known theorems. We then show that in the complex case, the only additional freedom allowed is complex conjugation. Finally, we apply those results to characterize the Fourier transform between certain function spaces.
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Acknowledgements
The authors would like to thank Mikhail Sodin for several useful discussions, Bo’az Klartag for explaining to us the failure of our original method of zero sets in the C ∞ n-dimensional setting, and the referee for numerous useful remarks and references, which allowed us to improve our results, and helped to provide better structure and context for the article.
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Artstein-Avidan, S., Faifman, D., Milman, V. (2012). On Multiplicative Maps of Continuous and Smooth Functions. In: Klartag, B., Mendelson, S., Milman, V. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2050. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29849-3_3
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DOI: https://doi.org/10.1007/978-3-642-29849-3_3
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